If the area of the region $\mathrm{S}=\left\{(\mathrm{x}, \mathrm{y}) ; 2 \mathrm{y}-\mathrm{y}^2 \leq \mathrm{x}^2 \leq 2 \mathrm{y}, \mathrm{x} \geq \mathrm{y}\right\}$ is equal to $\frac{\mathrm{n}+2}{\mathrm{n}+1}-\frac{\pi}{\mathrm{n}-1}$, then the natural number n is equal to ________.
